OF SECONDARY MINORS OF A PERSYMMETRIC DETERMINANT. 



515 



(4) Deleting the final secondary minor from each of the determinants in (A) as last 

 written we obtain three determinants 



(h,l) (h,k) 

 (I, 1) (/,*) 



(/,, 1) (h,l) 

 (k, 1) (k, I) 



(1,1) (l,h) 



and find at a glance that the first is still equal to the sum of the two last. It thus 

 appears that the latter simple identity, which we recognise to be the first case of 

 Kronecker's relation between n-line minors of a 2w-line axi symmetric determinant, 

 has an Extensional, viz., (A). 



(5) Before proceeding to the second generalisation it is desirable to state a few 

 elementary properties of persym metric determinants. These are : — 



(a) If the first row and last column, or first column and last roiv, of an n-line 

 persymmetric determinant P(a x , a 2 , ... , a 2n _ 1 ) he deleted, the resulting primary 

 minor is per symmetric, viz., P(a 2 , a 3 , ... , a 2n _ 2 ). 



(b) If the first row and first column be deleted the resulting primary minor is 

 persymmetric, viz., P(a 3 , a 4 , . . . , a 2n _ 1 ). 



(c) If the last roiv and last column be deleted, the resulting primary minor is 

 'persymmetric, viz., P(a x , a 2 , . . . , a 2n _ 3 ). 



(d) If any one of the theorems (a) , (b) , (c) be applied to any one of the persym- 

 metric determinants thus resulting, secondary minors will be obtained which are 

 persymmetric. 



(e) If a , /3 , . . . , \ be 2m — 1 numbers in equidifferent progression chosen from 

 1 , 2 , ... , 2n — 1 , then P(a a , a, p , . . . , a, K ) is an m-line persymmetric minor of 



M a l ' a 2 ' * • • ' a 2n-l)- 



(f) If any rows and the corresponding columns be deleted from a persymmetric 



minor of P(a : , a 2 , ... , a 2n _x) the 7'esult is an axisymmetric minor of the latter. 



For example, 



I 1, h + 1 , k+1, l+l j 

 ■ h, , k , I , n 



for i ] being persymmetric, if we delete its h th , k th , I th rows, which 



(k + l) th , (l+l) th rows of the original, and its /^ th , k th , I th columns, 

 which bear the same numbers in the original, the result will be axisymmetric. The 

 numbers of all the rows retained in the result, it may be noted, are each greater by 1 

 than the numbers of the columns. 



is axisymmetric ; 

 are the (h+l) th , 



(6) With these before us the second general theorem is readily dealt with. It is — 

 Secondary minors of any persymmetric determinant of order higher than the third 

 are connected by the relation 



k + 1 



I 



l + l 



k 



h + 1 

 I 



(B) 



where I > k > h > 



